Summary: This paper, following the purpose of synthesis of observes for nonlinear systems, investigates the question of equivalence of a nonlinear system with a bilinear system, or, more generally, a linear time-dependent system, plus an output injection. In a previous work by the same authors, such questions have already been dealt with from the local point of view. The goal herein is to examine the global situation. Using basic facts from algebraic topology, it is shown that in the single output case, whenever the possibility to bilinearize up to output injection holds locally everywhere, it also holds globally.